RBC Model with Indivisible Labor Kaiji Chen University of Oslo October 15, 2007 1 Last Class • Data on labor supply-fluctuations mostly driven by employment fluctuations • Effect of wage rate on labor supply — substitution effect: magnitude depend on Frisch elasticity — income effect: magnitude depending on how persistent is the wage change. • Effect of interest rate on labor supply • Performance of standard business cycle models 2 Road map • Model of Indivisible Labor • RBC Model with indivisible labor (and full employment insurance) • Aggregate and individual elasticity of labor supply 3 1. RBC MODEL WITH INDIVISIBLE LABOR 1 RBC Model with Indivisible Labor Indivisible labor • There is a continuum of ex-ante identical agents. • Suppose the period utility function is given by u (ct; ht) = log (ct) + v (1 - ht) where the function v satisfies v0 > 0, v" < 0. • Assume that household can either work full time, ht = h, 0 < h < 1,or not at all ht = 0. That is, they either have a job requiring fixed number of working hours, or they don't. 4 1. RBC MODEL WITH INDIVISIBLE LABOR_ • Rationale for this assumption: in the U.S. economy, about two thirds of variations in hours worked comes from individuals moving into and out of unemployment, with only one third from variations in hours when employed. • However, European data displays greater variance in hours worked per worker than in the number of workers. 5 1. RBC MODEL WITH INDIVISIBLE LABOR Labor lottery • Now, assume each individual can pick a probability it that he is employed and works h hours in period t, it 2 [0,1] . (This will make individuals happier than in the case it = {0,1} .) — Since all agents are ex-ante the same, they will choose the same probability it. • Hence, it is also the fraction of agents that are employed each period. • A lottery determines who will be actually be unemployed at each period. • Individuals can insure each other against the contingency of unemployment. 6 1. RBC MODEL WITH INDIVISIBLE LABOR The social planning problem • Alternatively, we can let the social planner choose 7t£, the fraction of population to work at each period. • Assume that all people get picked to work h hours with the same probability, so 7t£ is also the probability that a particular agent get picked. • Denote Ht = 7Tfh as the hours worked per capita. • Assume that the social planner provides full insurance against being unemployed. 7 1. RBC MODEL WITH INDIVISIBLE LABOR • As part of the dynamic social planning problem, the social planner solves cmax7rt (log cit + v (l - + (1 - 7rt) (log cot + v (1)) , subject to ^tcit + (1 - ^t) cot = ct where ct is total per capita consumption, which is given at this stage, c1t (cot) is consumption of an employed (unemployed) agent. • The solution to the above problem is cit = cot = ct- 8 1. RBC MODEL WITH INDIVISIBLE LABOR • The current period expected utility becomes Eu (ct;ht) = tt£ (log ct + v (l - h)) + (1 - 7rt)(tog ct + v (1)) = log ct - 7rt (v (1) - v (l - h)) + v (1) • Ignoring constants added to the utility function, we can rewrite the effective utility function as Eu (ct, ht) = log ct - i/>Ht where ^ = (v (1) - v (l - /T)) /h > 0. 9 1. RBC MODEL WITH INDIVISIBLE LABOR • Therefore, the decision variables for the social planner are the same as for a divisible labor model {ct,Ht,kt+1}t=0 ^=0 subject to fct+i + ct = (1 - 6) kt + eZtk"H^~a (1) ct > 0,Ht 2 [0; 1] and k0 given (2) zt = pzt-1 + "t (3) 10 1. RBC MODEL WITH INDIVISIBLE LABOR Key assumptions and implications • Key Assumptions — Full unemployment insurance. — All agents are ex-ante homogeneous. • Fluctuations in labor input comes from fluctuations in employment rather than fluctuations in hours per employed worker. • At the aggregate level, the Frisch elasticity of labor supply is infinity, though for a continuously employed worker, hours worked are constant (implying a value of 0 for Frisch elasticity of labor supply at the micro level). 11 1. RBC MODEL WITH INDIVISIBLE LABOR A simple example • Agents live for two periods, don't discount the future and only value consumption in the second period. — Abstract from capital accumulation, but let households store output between the first and the second period. • Let A\ and A2 denote labor productivity in the first and second period (which is also the wage rate in this case). • Social planner's problem max log C2 — Vhi — s.t. C2 = Aihi + A2hh2 12 1. RBC MODEL WITH INDIVISIBLE LABOR • Optimal Choice: if A\ > A?, then the agent should worked only in period 1. Vise Versa. • Extra unit of work brings about an extra disutility of work no matter when it is done. Thus should always work that extra unit in the period when labor is more productive. • In this case, h? = 0, hi = ^ and C2 = A. • For linear disutility of labor, effects of changes in productivity on labor supply are very strong. 13 1. RBC MODEL WITH INDIVISIBLE LABOR • Suppose the utility is log C2 + V> log (1 - hi) + V> log (1 - • Optimality condition A2 _ 1 - hi A = 1 - h2 • Labor supply does not respond as drastically to difference in Ai and A.2-As long as ^ not too big, work in both period. Small changes in ^ do not lead to drastic labor supply responses. • Data: labor input varies substantially over cycle, real wages only moderately. Linear specification more successful. 14 1. RBC MODEL WITH INDIVISIBLE LABOR_ Calibration of the RBC model with indivisible labor • pick so that the model economy reproduce amount of work equal to long run average in the data (about one third of their non-sleeping time ). — Compute the stead state hours worked. — Back out ip such that H = 3. • The calibration of other parameters are the same as before. 15 1. RBC MODEL WITH INDIVISIBLE LABOR 1.1 Quantitative Properties of the model Impulse response functions • Assume that before period 0, the economy is in the steady state. • At period 0, there is a positive technology shock, in the size of one standard deviation, that is zo = log (rai) • The reservation wage is rai w = -h (exp (^iA) - 1) h"!"* ■ ■ ■ ■ ■ ■ ■ where A = i_* is a constant, independent of individual characteristics. 27 2. MORE ON LABOR SUPPLY ELASTICITY • Workers with high //i and larger a demand a higher reservation wage. 28 2. MORE ON LABOR SUPPLY ELASTICITY Aggregate labor supply and the reservation wage distribution-an example • Suppose equal numbers of two types of workers exits in the economy, with reservation wage $10 and $20. • Suppose labor is indivisible. • At a wage rate of $10 and $20, the aggregate labor supply elasticity is infinite. otherwise, it is zero. • Intuition: whenever a mass in the reservation wage distribution exists, the aggregate labor supply elasticity can take a large value. 29 2. MORE ON LABOR SUPPLY ELASTICITY_ Aggregate labor supply and the reservation wage distribution-in general • Suppose many types of workers exit and that a worker works h hours if the market wage, w, exceeds the reservation wage. , , x [ h if w > w h (w) = < 0 otherwise • The reservation wage follows a distribution, where 0 (we) is the probability density function of the reservation wage. 30 2. MORE ON LABOR SUPPLY ELASTICITY • The aggregate labor supply function, H (w), is 'W r0 r w _ _ H (w) = / h( (we) dW = 0 (w) h 0 The aggregate labor supply elasticity T (w) = ^(^^ is 00 (w) w r(w) = 0 (w) • The aggregate elasticity depends on the concentration of workers: the marginal density, 00 (w) , relative to the cumulative density, 0 (w). • In the two-type workforce example, the aggregate elasticity is infinite where there is a mass of workers and zero elsewhere. 31 2. MORE ON LABOR SUPPLY ELASTICITY • In the lottery economy, the reservation wage distribution is degenerate (as the individuals are identical) at the equilibrium wage rate (00 (w) = oo) and the aggregate elasticity becomes infnity. 32